#4600661

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,9,10)(7,8,12,11)(13,14,18,20)(15,16,17,19)(22,26)(24,31)(27,33)(29,32);(6,10)(7,12)(13,15,18,17)(14,19,20,16)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,34)(32,36);(6,10)(8,11)(14,20)(15,17)(21,23,25,30)(22,27,26,33)(24,32,31,29)(28,36,34,35);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,19)(14,15,20,17)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,34)(32,36);(5,8,9,11)(6,7,10,12)(13,17,18,15)(14,19,20,16)(21,24,25,31)(22,28,26,34)(23,29,30,32)(27,35,33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4507857);